Prehomogeneous modules of commutative linear algebraic groups

Abstract

Let $A$ be a finite dimensional commutative associative algebra with unit over an algebraically closed field of characteristic zero. The group $G(A)$ of invertible elements is open in $A$ and thus $A$ has a structure of a prehomogeneous $G(A)$-module. We show that every prehomogeneous module of a commutative linear algebraic group appears this way. In particular, the number of equivalence classes of prehomogeneous $G$-modules is finite if and only if the corank of $G$ is at most $5$.